# Gauging, Measuring, and Controlling Critic Complexity in Actor-Critic Reinforcement Learning

arXiv: https://arxiv.org/abs/2607.00452 · HTML: https://mlsanity.com/paper/2607.00452/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 10 of its 12 references that are in the map.
- First posted on arXiv 1 Jul 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 10 of 10)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Deep Reinforcement Learning with Double Q-learning](https://mlsanity.com/paper/1509.06461/) (2015)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [Complexity Control Facilitates Reasoning-Based Compositional Generalization in Transformers](https://mlsanity.com/paper/2501.08537/) (2025)
- [Neural Weight Norm = Kolmogorov Complexity](https://mlsanity.com/paper/2605.10878/) (2026)

## Abstract (arXiv)

Actor-critic methods depend on learned critics, but critic quality is often evaluated only indirectly through return, temporal-difference error, or value loss. Critic complexity is introduced as an additional diagnostic and intervention dimension for actor-critic reinforcement learning. The analysis uses spectral effective-rank entropy, a rank-like summary of the singular-value distributions of critic weight matrices, to assess critic model complexity. Across TD3 and PPO experiments, critic complexity is tracked together with return and Monte Carlo value-estimation bias. The results show that critic complexity is measurable throughout training and is systematically associated with training behavior, while also making clear that the relationship is heterogeneous across algorithms, tasks, and hyperparameters. A direct complexity-control intervention is then evaluated by adding a spectral-entropy penalty to the critic loss. This intervention reliably changes the targeted spectral quantity, demonstrating that critic complexity can be controlled rather than only observed. Return effects are treated as task-dependent evidence rather than as a general performance claim, because overall complexity-control results vary.